-162. POPULATION EXPLOSION!

In 2008 Depedete’s population was approximately 800,000 people. If the population of Depedete grows at a rate of 5% per year, then Depedete can expect to increase by 40,000 people in 2009. In 2010, the increase should be 5% of its new population (840,000), which is 42,000 . Therefore, each year as the population changes, the rate of change in people per year, [tex]\frac{dP}{dt}[/tex], changes.

a. Write an equation that represents [tex]\frac{dP}{dt}[/tex], the rate of change of the population with respect to time.

b. Study the slope field at right for Depedete. The slope of each tangent line represents the rate of growth for P. Examine the tangent lines for P=800,000. Why do they all have the same slope?

c. Place your paper over the slope field. If P(0)=800,000, draw the particular solution for P given this initial condition. What type of function is P(t) ?

d. Use implicit integration to write an equation for P(t).

e. The slopes of P are not the same for each value of t, yet depend only on the values of t. Explain why. Hint: Think about the role of the constant of integration in this problem versus other problems.

f. Write equation that will estimate future populations of Depedete if the city grows at a rate of 3.5% per year. Use this equation, and the fact that the 2008 population was 800,000 to estimate the population in the year 2099 .